Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Entropy

Giovanni Pistone¹  Bertrand Lods² 

[1] Castro Statistics, Collegio Carlo Alberto, Via Real Collegio 30, 10024 Moncalieri, Italy;Department of Economics and Statistics, Università di Torino & Collegio Carlo Alberto, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy;

关键词: information geometry;    Orlicz space;    spatially homogeneous Boltzmann equation;    Kullback-Leibler divergence;    Hyvärinen divergence;   

DOI  :  10.3390/e17064323

来源: DOAJ

【 摘 要 】

Information Geometry generalizes to infinite dimension by modeling the tangentspace of the relevant manifold of probability densities with exponential Orlicz spaces.We review here several properties of the exponential manifold on a suitable set Ɛ ofmutually absolutely continuous densities. We study in particular the fine properties of theKullback-Liebler divergence in this context. We also show that this setting is well-suitedfor the study of the spatially homogeneous Boltzmann equation if Ɛ is a set of positivedensities with finite relative entropy with respect to the Maxwell density. More precisely,we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell’sweak form as a composition of elementary operations in the exponential manifold, namelytensor product, conditioning, marginalization and we prove in a geometric way the basicfacts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing,besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. Thisrequires us to generalize our approach to Orlicz–Sobolev spaces to include derivatives.

【 授权许可】

Unknown